To solve the expression [tex]\(\frac{2}{3} + \frac{1}{4} \times \frac{2}{5} - \frac{3}{4} + \frac{5}{2}\)[/tex], we will follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
Let's break it down step by step:
1. Multiplication:
[tex]\[
\frac{1}{4} \times \frac{2}{5} = \frac{1 \times 2}{4 \times 5} = \frac{2}{20} = \frac{1}{10}
\][/tex]
2. Expression Update:
[tex]\[
\frac{2}{3} + \frac{1}{10} - \frac{3}{4} + \frac{5}{2}
\][/tex]
3. Finding a common denominator:
To simplify the addition and subtraction, we will find a common denominator for [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{1}{10}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], and [tex]\(\frac{5}{2}\)[/tex].
- The least common multiple (LCM) of 3, 10, 4, and 2 is 60.
- Convert each fraction to have a denominator of 60:
- [tex]\(\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}\)[/tex]
- [tex]\(\frac{1}{10} = \frac{1 \times 6}{10 \times 6} = \frac{6}{60}\)[/tex]
- [tex]\(\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}\)[/tex]
- [tex]\(\frac{5}{2} = \frac{5 \times 30}{2 \times 30} = \frac{150}{60}\)[/tex]
4. Updating Expression with Common Denominator:
[tex]\[
\frac{40}{60} + \frac{6}{60} - \frac{45}{60} + \frac{150}{60}
\][/tex]
5. Combining the Fractions:
[tex]\[
\frac{40 + 6 - 45 + 150}{60} = \frac{151}{60}
\][/tex]
The simplification gives us the result [tex]\(\frac{151}{60}\)[/tex], which corresponds to the given result.
None of the options A, B, C, or D match the obtained fraction. Therefore, it appears the correct numerical answer, based on detailed step-by-step calculation, is [tex]\(\frac{151}{60}\)[/tex], which is not among the given options.
